Integral geometry and geometric probability:
Gespeichert in:
| Vorheriger Titel: | Integral geometry and geometric probability |
|---|---|
| 1. Verfasser: | |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
2009
|
| Ausgabe: | 2. ed., [digital printing] |
| Schriftenreihe: | Cambridge mathematical library
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Originally published: Reading, Mass.; London, Addison-Wesley, 1976 |
| Beschreibung: | XVII, 404 S. graph. Darst. |
| ISBN: | 9780521523448 |
Internformat
MARC
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| 020 | |a 9780521523448 |c Pbk |9 978-0-521-52344-8 | ||
| 035 | |a (OCoLC)634645039 | ||
| 035 | |a (DE-599)BSZ310803497 | ||
| 040 | |a DE-604 |b ger | ||
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| 100 | 1 | |a Santaló, Luis A. |d 1911-2001 |e Verfasser |0 (DE-588)127957278 |4 aut | |
| 245 | 1 | 0 | |a Integral geometry and geometric probability |c Luis A. Santaló |
| 250 | |a 2. ed., [digital printing] | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2009 | |
| 300 | |a XVII, 404 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge mathematical library | |
| 500 | |a Originally published: Reading, Mass.; London, Addison-Wesley, 1976 | ||
| 650 | 4 | |a Integral geometry | |
| 650 | 4 | |a Geometric probabilities | |
| 650 | 0 | 7 | |a Geometrische Wahrscheinlichkeit |0 (DE-588)4156727-4 |2 gnd |9 rswk-swf |
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| 650 | 0 | 7 | |a Wahrscheinlichkeitsrechnung |0 (DE-588)4064324-4 |2 gnd |9 rswk-swf |
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| 689 | 3 | |8 1\p |5 DE-604 | |
| 780 | 0 | 0 | |i 1. ed. u.d.T. |t Integral geometry and geometric probability |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018650681&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-018650681 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804140732208906240 |
|---|---|
| adam_text | Contents
Editor s Statement
....................... xi
Foreword
........................... xiii
Preface
............................ xv
Part I: INTEGRAL GEOMETRY IN THE PLANE
Chapter
1.
Convex Sets in the Plane
............... 1
1. Introduction
.................... 1
2.
Envelope of a Family of Lines
............ 2
3.
Mixed Areas of Minkowski
.............. 4
4.
Some Special Convex Sets
.............. 6
5.
Surface Area of the Unit Sphere and Volume of the Unit
Ball
........................ 9
6.
Notes and Exercises
................. 10
Chapter
2.
Sets of Points and
Poisson
Processes in the Plane
.... 12
1.
Density for Sets of Points
.............. 12
2.
First Integral Formulas
............... 13
3.
Sets of Triples of Points
............... 16
4.
Homogeneous Planar
Poisson
Point Processes
..... 17
5.
Notes
....................... 20
Chapters. Sets of Lines in the Plane
............... 27
1. Density for Sets of Lines
................ 27
2.
Lines That Intersect a Convex Set or a Curve
..... 30
3.
Lines That Cut or Separate Two Convex Sets
..... 32
4.
Geometric Applications
............... 34
5.
Notes and Exercises
................. 37
Chapter
4.
Pairs of Points and Pairs of Lines
........... 45
1.
Density for Pairs of Points
.............. 45
V
Contents
2. Integrals
for the
Power
of the Chords of a Convex Set
. . 46
3.
Density for Pairs of Lines
.............. 49
4.
Division of the Plane by Random Lines
........ 51
5.
Notes
.......................
58
Chapter
5.
Sets of Strips in the Plane
68
1.
Density for Sets of Strips
.............. 68
2.
Buffon s Needle Problem
............... 71
3.
Sets of Points, Lines, and Strips
............ 72
4.
Some Mean Values
................. 75
5.
Notes
....................... 77
Chapter
6.
The Group of Motions in the Plane: Kinematic Density
. . 80
1.
The Group of Motions in the Plane
.......... 80
2.
The Differential Forms on
Sil
............. 82
3.
The Kinematic Density
............... 85
4.
Sets of Segments
.................. 89
5.
Convex Sets That Intersect Another Convex Set
.... 93
6.
Some Integral Formulas
............... 95
7.
A Mean Value; Coverage Problems
.......... 98
8.
Notes and Exercises
................. 100
Chapter
7.
Fundamental Formulas of
Poincaré
and Blaschke
..... 109
1.
A New Expression for the Kinematic Density
...... 109
2.
Poincaré s
Formula
................. 1
H
3.
Total Curvature of a Closed Curve and of a Plane Domain
112
4.
Fundamental Formula of Blaschke
..........
I·3
5.
The Isoperimetric Inequality
............. 49
6.
Hadwiger s Conditions for a Domain to Be Able to Contain
Another
...................... 121
7.
Notes
....................... 123
Chapters. Lattices of Figures
.................. 128
I- Definitions and Fundamental Formula
.........
l28
2.
Lattices of Domains
................. 131
3.
Lattices of Curves
................. 134
4.
Lattices of Points
.................. 135
5.
Notes and Exercise
... ..... 138
Contents
Part II.
GENERAL INTEGRAL
GEOMETRY
Chapter
9. Differential
Forms and Lie Groups
...........143
1. Differential Forms
.................143
2.
Pfaffian Differential Systems
.............145
3.
Mappings of Differentiable Manifolds
.........148
4.
Lie Groups; Left and Right Translations
........149
5.
Left-Invariant Differential Forms
...........150
6.
Maurer-Cartan Equations
..............152
7.
Invariant Volume Elements of a Group: Unimodular
Groups
......................156
8.
Notes and Exercises
.................160
Chapter
10.
Density and Measure in Homogeneous Spaces
......165
1.
Introduction
....................165
2.
Invariant Subgroups and Quotient Groups
.......169
3.
Other Conditions for the Existence of a Density on Homo¬
geneous Spaces
...................
1
70
4.
Examples
.....................171
5.
Lie Transformation Groups
.............173
6.
Notes and Exercises
.................175
Chapter
11.
The
Affine
Groups
.................179
1.
The Groups of
Affine
Transformations
........179
2.
Densities for Linear Spaces with Respect to Special Homo¬
geneous Affinities
..................183
3.
Densities for Linear Subspaces with Respect to the Special
Nonhomogeneous
Affine
Group
...........187
4.
Notes and Exercises
.................189
Chapter
12.
The Group of Motions in
£„.............196
1. Introduction
....................196
2.
Densities for Linear Spaces in
E„...........199
3.
A Differential Formula
...............200
4.
Density for r-Planes about a Fixed q-Plane
.......201
5.
Another Form of the Density for r-Planes in
£„.....204
6.
Sets of Pairs of Linear Spaces
............205
7.
Notes
.......................207
Contents
Part
III. INTEGRAL
GEOMETRY IN
£„
Chapter
13.
Convex Sets in
£„..................215
1. Convex Sets and
Quermassintegrale..........215
2.
Cauchy s Formula
.................217
3.
Parallel Convex Sets;
Steiner
s
Formula
........220
4.
Integral Formulas Relating to the Projections of a Convex
Set on Linear Subspaces
...............221
5.
Integrals of Mean Curvature
.............222
6.
Integrals of Mean Curvature and
Quermassintegrale . . . 224
7.
Integrals of Mean Curvature of a Flattened Convex Body
227
8.
Notes
.......................229
Chapter
14.
Linear Subspaces, Convex Sets, and Compact Manifolds
. 233
1. Sets of r-Planes That Intersect a Convex Set
...... 233
2.
Geometric Probabilities
............... 235
3.
Crofton s Formulas in
£„.............. 237
4.
Some Relations between Densities of Linear Subspaces
. . 240
5.
Linear Subspaces That Intersect a Manifold
...... 243
6.
Hypersurfaces and Linear Spaces
........... 247
7.
Notes
....................... 248
Chapter
15.
The Kinematic Density in
£„.............256
1. Formulas on Densities
................ 256
2.
Integral of the Volume
ffr + ł_„(M
η Μ )
........ 258
3.
A Differential Formula
............... 260
4.
The Kinematic Fundamental Formula
......... 262
5.
Fundamental Formula for Convex Sets
........ 267
6.
Mean Values for the Integrals of Mean Curvature
.... 267
7.
Fundamental Formula for Cylinders
......... 270
8.
Some Mean Values
................. 272
9.
Lattices in
£„.................... 274
10.
Notes and Exercise
................. 275
Chapter
16.
Geometric and Statistical Applications; Stereology
.... 282
I· Size Distribution of Particles Derived from the Size
Distribution of Their Sections
............ 282
2.
Intersection with Random Planes
........... 286
3.
Intersection with Random Lines
........... 289
4.
Notes
........ 290
Contents
Part IV.
INTEGRAL
GEOMETRY IN
SPACES
OF
CONSTANT CURVATURE
Chapter
17.
Noneuclidean Integral Geometry
............ 299
1.
The «-Dimensional Noneuclidean Space
........ 299
2.
The Gauss-Bonnet Formula for Noneuclidean Spaces
. . 302
3.
Kinematic Density and Density for r-Planes
...... 304
4.
Sets of r-Planes That Meet a Fixed Body
........ 309
5.
Notes
....................... 311
Chapter
18.
Crofton s Formulas and the Kinematic Fundamental Formula
in Noneuclidean Spaces
................ 316
1.
Crofton s Formulas
................. 316
2.
Dual Formulas in Elliptic Space
........... 318
3.
The Kinematic Fundamental Formula in Noneuclidean
Spaces
....................... 320
4.
Steiner s Formula in Noneuclidean Spaces
....... 321
5.
An Integral Formula for Convex Bodies in Elliptic Space
322
6.
Notes
....................... 323
Chapter
19.
Integral Geometry and Foliated Spaces; Trends in Integral
Geometry
..................... 330
1.
Foliated Spaces
................... 330
2.
Sets of Geodesies in a Riemann Manifold
....... 331
3.
Measure of Two-Dimensional Sets of Geodesies
..... 334
4.
Measure of (2n
-
2)-Dimensional Sets of Geodesies
. . . 336
5.
Sets of Geodesic Segments
.............. 338
6.
Integral Geometry on Complex Spaces
........ 338
7.
Symplectic Integral Geometry
............ 344
8.
The Integral Geometry of Gelfand
.......... 345
9.
Notes
....................... 349
Appendix. Differential Forms and Exterior Calculus
........ 353
1.
Differential Forms and Exterior Product
........ 353
2.
Two Applications of the Exterior Product
....... 357
3.
Exterior Differentiation
............... 358
4.
Stokes Formula
.................. 359
5.
Comparison with Vector Calculus in Euclidean Three-
Dimensional Space
................. 360
6.
Differential Forms over Manifolds
.......... 361
Bibliography and References
................... 363
Author Index
......................... 395
Subject Index
......................... 399
|
| any_adam_object | 1 |
| author | Santaló, Luis A. 1911-2001 |
| author_GND | (DE-588)127957278 |
| author_facet | Santaló, Luis A. 1911-2001 |
| author_role | aut |
| author_sort | Santaló, Luis A. 1911-2001 |
| author_variant | l a s la las |
| building | Verbundindex |
| bvnumber | BV035791349 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)634645039 (DE-599)BSZ310803497 |
| discipline | Mathematik |
| edition | 2. ed., [digital printing] |
| format | Book |
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| id | DE-604.BV035791349 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:04:39Z |
| institution | BVB |
| isbn | 9780521523448 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018650681 |
| oclc_num | 634645039 |
| open_access_boolean | |
| owner | DE-703 DE-20 |
| owner_facet | DE-703 DE-20 |
| physical | XVII, 404 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | Cambridge mathematical library |
| spelling | Santaló, Luis A. 1911-2001 Verfasser (DE-588)127957278 aut Integral geometry and geometric probability Luis A. Santaló 2. ed., [digital printing] Cambridge [u.a.] Cambridge University Press 2009 XVII, 404 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical library Originally published: Reading, Mass.; London, Addison-Wesley, 1976 Integral geometry Geometric probabilities Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd rswk-swf Integralgeometrie (DE-588)4161911-0 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Geometrie (DE-588)4020236-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Integralgeometrie (DE-588)4161911-0 s Geometrische Wahrscheinlichkeit (DE-588)4156727-4 s 1\p DE-604 1. ed. u.d.T. Integral geometry and geometric probability Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018650681&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Santaló, Luis A. 1911-2001 Integral geometry and geometric probability Integral geometry Geometric probabilities Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Integralgeometrie (DE-588)4161911-0 gnd Geometrie (DE-588)4020236-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4156727-4 (DE-588)4161911-0 (DE-588)4020236-7 (DE-588)4064324-4 |
| title | Integral geometry and geometric probability |
| title_auth | Integral geometry and geometric probability |
| title_exact_search | Integral geometry and geometric probability |
| title_full | Integral geometry and geometric probability Luis A. Santaló |
| title_fullStr | Integral geometry and geometric probability Luis A. Santaló |
| title_full_unstemmed | Integral geometry and geometric probability Luis A. Santaló |
| title_old | Integral geometry and geometric probability |
| title_short | Integral geometry and geometric probability |
| title_sort | integral geometry and geometric probability |
| topic | Integral geometry Geometric probabilities Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Integralgeometrie (DE-588)4161911-0 gnd Geometrie (DE-588)4020236-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Integral geometry Geometric probabilities Geometrische Wahrscheinlichkeit Integralgeometrie Geometrie Wahrscheinlichkeitsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018650681&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT santaloluisa integralgeometryandgeometricprobability |